Measuring effects of categorical factors on binomial outcome with many groups I'd like to do some analysis of shooting efficiency in basketball when a team is leading (AHEAD) or trailing (BEHIND) by less than 8 points and whether they are HOME or AWAY. Here are a few examples of the data:
Ray Allen   HOME    BEHIND  59.4%   134
Ray Allen   HOME    AHEAD   57.13%  132
Ray Allen   AWAY    BEHIND  49.1%   166
Ray Allen   AWAY    AHEAD   48.03%  126
Jason Terry AWAY    BEHIND  56.6%   242
Jason Terry HOME    BEHIND  52.0%   193
Jason Terry AWAY    AHEAD   50.05%  198
Jason Terry HOME    AHEAD   48.73%  207
Jamal Crawford  AWAY    AHEAD   51.65%  82
Jamal Crawford  HOME    AHEAD   42.50%  178
Jamal Crawford  AWAY    BEHIND  35.5%   129
Jamal Crawford  HOME    BEHIND  33.4%   118
Kevin Durant    HOME    BEHIND  48.6%   222
Kevin Durant    HOME    AHEAD   44.05%  248
Kevin Durant    AWAY    BEHIND  41.4%   325
Kevin Durant    AWAY    AHEAD   40.07%  213

The 4th column is the FG% (i.e. proportion of made shots) and the 5th column is the number of shots (i.e. trials).
You can see even with these 4 players (and there are roughly 200 in the data set), that there is variation of the mean FG% between players, and for each player, there is not a consistent pattern in whether they are "better" at HOME or AWAY or AHEAD or BEHIND. So there's a lot of variance between groups and within groups as far as I can tell.
I thought about using lmer, but I wasn't sure how to do that for this problem, because if I just use the FG% as the outcome, I lose the information about how many shots were taken. Eventually, I'd like to put this into BUGS, but I thought there might be a more straightforward way for now, because I'm not quite ready for that yet.
I should just add that what I'm really after is a way to determine whether a player is "really" better under one of these conditions, or are the apparent differences just due to noise/variation from small sample sizes.
Thanks for any advice.
 A: To circumvent the 200-players problem, you could fit whichever model you choose (logit, binomial...), without the player variable as such, but inside a discrete mixture framework. You'll have to process the data right (for instance you want to make sure that all stats of a single player are taken together, and you'll have to determine the optimal number of clusters in the mixture) but the fitted mixture model will group players into clusters, which should reflect differences in performance, or rather differences in how the conditions (home and ahead) affect performance. This is very easy and fast with R package flexmix.
Building on the same idea, you could also just run an unsupervized clustering algo (k-means, gaussian mixture, self-organizing map) on the data transformed as such: each player has one vector of 8 values $(rate_{home,lead}, N_{home, lead}, rate_{home, behind}, N_{home, behind}, ...)$. In that case each player belongs to a cluster of players with similar characteristics, and you can check whether the differences between clusters are significant.
A: I think you could fit a logistic regression model using player, ahead/behind, home/away, percentage success and number of shots taken under those conditions as possible covariates.  Then difficulty with player is that you have over 200.  I think that success percentage under specific conditions could serve as a substitute for player since the player and his past performance under the conditions should be highly related to the outcome.  To predict for individual players you just use that player's other covariates.
